Fractional integral associated to Schrödinger operator on the Heisenberg groups in central generalized Morrey spaces
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Authors
Ahmet Eroglu
- Department of Mathematics, Nigde Omer Halisdemir University, Nigde, Turkey.
Tahir Gadjiev
- Institute of Mathematics and Mechanics, NAS of Azerbaijan, AZ1141 Baku, Azerbaijan.
Faig Namazov
- Baku State University, AZ1141 Baku, Azerbaijan.
Abstract
Let \(L=-\Delta_{\mathbb{H}_n}+V\) be a Schrödinger operator on the Heisenberg groups \(\mathbb{H}_n\), where the non-negative potential \(V\) belongs to the reverse Hölder class \(RH_{Q/2}\)
and \(Q\) is the homogeneous dimension of \(\mathbb{H}_n\). Let \(b\) belong to a new \(BMO_{\theta}(\mathbb{H}_n,\rho)\) space, and let \({\cal I}_{\beta}^{L}\) be the fractional integral operator associated with \(L\).
In this paper, we study the boundedness of the operator \({\cal I}_{\beta}^{L}\) and its commutators \([b,{\cal I}_{\beta}^{L}]\) with \(b \in BMO_{\theta}(\mathbb{H}_n,\rho)\)
on central generalized Morrey spaces \(LM_{p,\varphi}^{\alpha,V}(\mathbb{H}_n)\) and generalized Morrey spaces \(M_{p,\varphi}^{\alpha,V}(\mathbb{H}_n)\) associated with Schrödinger operator.
We find the sufficient conditions on the pair \((\varphi_1,\varphi_2)\) which ensures the boundedness of the operator \({\cal I}_{\beta}^{L}\)
from \(LM_{p,\varphi_1}^{\alpha,V}(\mathbb{H}_n)\) to \(LM_{q,\varphi_2}^{\alpha,V}(\mathbb{H}_n)\) and from \(M_{p,\varphi_1}^{\alpha,V}(\mathbb{H}_n)\) to \(M_{q,\varphi_2}^{\alpha,V}(\mathbb{H}_n)\), \(1/p-1/q=\beta/Q\).
When \(b\) belongs to \(BMO_{\theta}(\mathbb{H}_n,\rho)\) and \((\varphi_1,\varphi_2)\) satisfies some conditions, we also show that the commutator operator \([b,{\cal I}_{\beta}^{L}]\) is bounded
from \(LM_{p,\varphi_1}^{\alpha,V}(\mathbb{H}_n)\) to \(LM_{q,\varphi_2}^{\alpha,V}(\mathbb{H}_n)\) and from \(M_{p,\varphi_1}^{\alpha,V}\) to \(M_{q,\varphi_2}^{\alpha,V}\), \(1/p-1/q=\beta/Q\).
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ISRP Style
Ahmet Eroglu, Tahir Gadjiev, Faig Namazov, Fractional integral associated to Schrödinger operator on the Heisenberg groups in central generalized Morrey spaces, Journal of Nonlinear Sciences and Applications, 11 (2018), no. 8, 984--993
AMA Style
Eroglu Ahmet, Gadjiev Tahir, Namazov Faig, Fractional integral associated to Schrödinger operator on the Heisenberg groups in central generalized Morrey spaces. J. Nonlinear Sci. Appl. (2018); 11(8):984--993
Chicago/Turabian Style
Eroglu, Ahmet, Gadjiev, Tahir, Namazov, Faig. "Fractional integral associated to Schrödinger operator on the Heisenberg groups in central generalized Morrey spaces." Journal of Nonlinear Sciences and Applications, 11, no. 8 (2018): 984--993
Keywords
- Schrödinger operator
- Heisenberg group
- central generalized Morrey space
- fractional integral
- commutator
- BMO
MSC
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